Through a given point to draw a straight line parallel
to a given straight line.

Let A be the given point, and
BC the given straight
line;
thus it is required to draw through the
point A a
straight line parallel to the straight line
BC.

Let a point D be taken at
random on BC, and let
AD be joined;
on the straight line DA,
and at the point A on it,
let the angle DAE be constructed
equal to the
angle ADC
[I. 23]
;
and let the straight line AF
be produced in a straight line with EA.

ABCDEF

Then, since the straight line AD
falling on the two straight lines
BC, EF
has made the alternate angles
EAD, ADC
equal to one another,
therefore EAF is parallel
to BC.
[I. 27]

Therefore through the given point
A the straight
line EAF
has been drawn parallel to the given straight
line BC.
Q.E.F.